They are nicknames for algorithm updates Google has done. It's not really I portant for someone starting out, but it has effected some people's rankings.

Here's a nice video that explains PANDA and what you need to do

How Google_s Panda Update Changed SEO Best Practices Forever - YouTube
Here is info on PENGUIN How to fix a Google Penguin Penalty by using Natural Anchor Text. - YouTube

Click THANKS if you like this

What Panda and Penguin mean, is that it is now even more important to have good quality content, and to backlink in a way that Google doesn't know you are doing it yourself, ie. you need a wide variety of DIFFERENT types of backlinks, not just lots of one or two types.

There are SO many questions on the WF about "which type of backlink is best", and the answer is, of course, that building a lot of any one type on its own will make it obvious to Google that you are building them yourself, and so they will give you less traffic. So learn about a lot of different types of backlinks, such as articles, document shares, video, video embeds, RSS feeds, social bookmarks, web2.0 pages, blog comments, forum profiles and signatures, and other types, and use them ALL.

Chris

In a nutshell, both are Google algorithms changes. Panda penalized sites with duplicate content and content farms while Penguin penalized sites that were over optimized for one or more keywords.

Google Panda : Earlier Update in Google Algorithm to De-ranked and penalize those website have use of Low Quality or crappy content .After the Update 'Content becoming King for any website ' A website having low Quality content or duplicate ,not allow to get placed with top Google Search ranking .
Google Penguin : Latest Update by Google ,recently in 24th April . The task is to penalize or De-ranked those spoils the meaning of SEO and play with Google as Web spammer. Over-optimization ,Hidden text ,Unnatural linking is the target this time .Sites that are over optimized and come in ranking by unnatural linking get De-ranked .
Both Updates demonstrate to provide Better Results in SERp's .

Whenever Google does a big update to its search engine, they give it a name. I don't know why. Maybe they just have too much time on their hands. When it comes to the search engines, all you have to do is provide unique, quality content that provides value to the searcher. This is, and will always be, the main goal of a search engine. If you do this, the updates will help you, and Google will give you more traffic.

Joey

This image may help you to understand G-panda and penguin.

Those are the most discussed topic for last few months in internet world. In this forum, you will find a lot of information regarding that.
However, these two are Google's algorithm update that works against copied/duplicate contents and bad quality links. If your site contains something like that, there is strong possibility that you will be penalized for that and will loss the rank. That will ultimately reduce the traffic in your site.